The full reasoning can be found below the Sudoku.
Jan 31 - Super Hard
Puzzle Copyright © Kevin Stone
www.brainbashers.com
Reasoning
R1C6 can only be <8>
R3C8 can only be <7>
R9C6 can only be <4>
R2C5 can only be <1>
R1C4 can only be <7>
R5C6 can only be <3>
R1C5 can only be <6>
R5C4 can only be <9>
R9C4 can only be <1>
R4C5 can only be <4>
R4C1 can only be <5>
R4C9 can only be <9>
R3C1 is the only square in row 3 that can be <6>
R5C1 is the only square in row 5 that can be <1>
R9C7 is the only square in column 7 that can be <7>
R7C1 is the only square in row 7 that can be <7>
Squares R7C2 and R8C1 in block 7 form a simple naked pair. These 2 squares both contain the 2 possibilities <24>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R8C3 - removing <4> from <456> leaving <56>
R9C1 - removing <2> from <238> leaving <38>
R9C2 - removing <2> from <2358> leaving <358>
R1C3 is the only square in column 3 that can be <4>
Squares R3C2<28>, R5C2<48> and R7C2<24> in column 2 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <248>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R1C2 - removing <2> from <235> leaving <35>
R9C2 - removing <8> from <358> leaving <35>
Squares R3C2 and R3C9 in row 3 and R7C2 and R7C9 in row 7 form a Simple X-Wing pattern on possibility <2>. All other instances of this possibility in columns 2 and 9 can be removed.
R1C9 - removing <2> from <1235> leaving <135>
R8C9 - removing <2> from <2456> leaving <456>
R9C9 - removing <2> from <256> leaving <56>
R9C5 is the only square in row 9 that can be <2>
R8C5 can only be <9>
R9C8 is the only square in row 9 that can be <9>
Squares R1C2<35>, R1C8<15> and R1C9<135> in row 1 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <135>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R1C1 - removing <3> from <239> leaving <29>
R1C7 - removing <5> from <259> leaving <29>
Squares R5C9<457>, R6C9<47>, R8C9<456> and R9C9<56> in column 9 form a comprehensive naked quad. These 4 squares can only contain the 4 possibilities <4567>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R1C9 - removing <5> from <135> leaving <13>
R2C9 - removing <5> from <358> leaving <38>
R7C9 - removing <4> from <124> leaving <12>
Squares R6C1 and R6C9 in row 6 and R8C1 and R8C9 in row 8 form a Simple X-Wing pattern on possibility <4>. All other instances of this possibility in columns 1 and 9 can be removed.
R5C9 - removing <4> from <457> leaving <57>
Squares R2C3 (XY), R1C2 (XZ) and R2C9 (YZ) form an XY-Wing pattern on <3>. All squares that are buddies of both the XZ and YZ squares cannot be <3>.
R2C1 - removing <3> from <389> leaving <89>
R1C9 - removing <3> from <13> leaving <1>
R1C8 can only be <5>
R7C9 can only be <2>
R7C2 can only be <4>
R3C9 can only be <8>
R8C7 can only be <5>
R8C3 can only be <6>
R2C7 can only be <9>
R9C9 can only be <6>
R8C9 can only be <4>
R1C2 can only be <3>
R5C8 can only be <4>
R2C1 can only be <8>
R1C7 can only be <2>
R3C2 can only be <2>
R2C9 can only be <3>
R5C2 can only be <8>
R7C8 can only be <1>
R6C9 can only be <7>
R6C5 can only be <8>
R5C9 can only be <5>
R8C1 can only be <2>
R1C1 can only be <9>
R9C2 can only be <5>
R2C3 can only be <5>
R6C1 can only be <4>
R9C1 can only be <3>
R9C3 can only be <8>
R5C5 can only be <7>
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